Bell States: Construction and Entanglement Verification

Hello! Welcome to your sixth lesson in the "Quantum Computing Foundations" module.

In our last lesson, we explored the process of quantum measurement. We saw how measuring a part of a larger, pure quantum system can lead to probabilistic outcomes, leaving the measured subsystem in a mixed state described by a density matrix. This phenomenon is a direct consequence of entanglement, the central topic of today's lesson.

Entanglement is a uniquely quantum resource that underpins much of the power of quantum computation. We will begin our formal study of it by focusing on its most fundamental manifestation: the Bell states.

The learning outcome for this lesson is to construct the Bell states and demonstrate their entangled nature by calculating their Schmidt decomposition and von Neumann entropy.

To achieve this, we will:

Construct the four Bell states using a simple quantum circuit composed of Hadamard and CNOT gates.
Introduce the Schmidt decomposition, a powerful mathematical tool for analyzing the structure of bipartite pure states.
Apply the Schmidt decomposition to the Bell states to prove they are entangled.
Use the von Neumann entropy of the reduced density matrix to quantify this entanglement and show that the Bell states are "maximally entangled".

1. Constructing the Bell States

The Bell states are a set of four specific two-qubit states that form an orthonormal basis, known as the Bell basis. They are the simplest and most important examples of entangled states. A standard way to generate them is to start with one of the four computational basis states $∣00 ⟩, ∣01 ⟩, ∣10 ⟩, ∣11 ⟩$ and apply a circuit consisting of a Hadamard gate on the first qubit followed by a CNOT gate.

The following video provides a detailed walkthrough of this process, including the matrix algebra involved.

Bell States from 2-Qubit Computational Basis States via Quantum Circuit (Hadamard and CNOT Gates)

This video, 'Bell States from 2-Qubit Computational Basis States via Quantum Circuit', meticulously explains how a Hadamard gate followed by a CNOT gate transforms the computational basis into the Bell basis.

Please watch the video, focusing on these key segments: Introduction (00:00 - 01:51): Understand the circuit diagram. Unitary Operator (01:51 - 08:49): See how the total unitary operator U is constructed from the tensor product of individual gate matrices. Pay attention to the ordering. Resultant Unitary (08:49 - 12:02): Observe the final 4x4 matrix for the entire circuit. Mapping to Bell States (14:57 - 19:08): This is the core part, showing how each computational basis state maps to a unique Bell state. Properties and Entanglement (19:08 - 22:49): Note the initial discussion on why these states cannot be factored.

As the video demonstrates, applying the unitary operator $U = CNOT \cdot (H \otimes I)$ to the computational basis states yields the four Bell states. Using the more standard $Φ, Ψ$ notation, these are:

$U ∣00 ⟩ = ∣ Φ^{+} ⟩ = \frac{1}{2} (∣00 ⟩ + ∣11 ⟩)$
$U ∣01 ⟩ = ∣ Ψ^{+} ⟩ = \frac{1}{2} (∣01 ⟩ + ∣10 ⟩)$
$U ∣10 ⟩ = ∣ Φ^{-} ⟩ = \frac{1}{2} (∣00 ⟩ - ∣11 ⟩)$
$U ∣11 ⟩ = ∣ Ψ^{-} ⟩ = \frac{1}{2} (∣01 ⟩ - ∣10 ⟩)$

The crucial property mentioned at the end of the video is that these states are entangled, meaning they cannot be written as a simple tensor product of two individual qubit states, i.e., $∣ ψ ⟩_{A} \otimes ∣ ϕ ⟩_{B}$ . Let's now introduce a formal tool to analyze this property.

2. The Schmidt Decomposition

The Schmidt decomposition is a theorem that provides a canonical representation for a pure state in a bipartite quantum system. It is immensely useful for determining whether a state is entangled and for quantifying that entanglement.

The following video gives an excellent overview of the concept.

The Schmidt Decomposition (Overview)

This video, 'The Schmidt Decomposition (Overview)', explains the purpose of the decomposition, how to interpret it, and its connection to entanglement via the Schmidt number.

Please watch the following segments: Introduction (00:00 - 01:03): Get a high-level view of what the decomposition is for. Two-Qubit System (01:03 - 03:59): Understand the form of the decomposition for a two-qubit state. Schmidt Number (05:52 - 06:45): This is a key concept. Understand how the number of terms indicates separability or entanglement. Entangled Example (06:45 - 09:05): See how the decomposition clarifies the correlations in an entangled state.

Let's formalize the statement of the theorem.

Schmidt Decomposition Theorem: For any pure state $∣ ψ ⟩$ of a composite system $H_{A} \otimes H_{B}$ , there exist orthonormal sets of states ${∣ u_{i} ⟩_{A}}$ in $H_{A}$ and ${∣ v_{i} ⟩_{B}}$ in $H_{B}$ such that:

∣ ψ ⟩ = i = 1 \sum k λ_{i} ∣ u_{i} ⟩_{A} \otimes ∣ v_{i} ⟩_{B}

where the $λ_{i}$ are non-negative real numbers called Schmidt coefficients, satisfying $\sum_{i} λ_{i}^{2} = 1$ . The number of non-zero coefficients, $k$ , is called the Schmidt rank of the state.

The Schmidt rank is a direct indicator of entanglement:

If $k = 1$ , the state is a product state (separable). The sum has only one term: $∣ ψ ⟩ = λ_{1} ∣ u_{1} ⟩_{A} \otimes ∣ v_{1} ⟩_{B}$ . Since $λ_{1} = 1$ , this is a simple product.
If $k > 1$ , the state is entangled.

The Schmidt decomposition is essentially the singular value decomposition (SVD) of the matrix of coefficients of the state vector when written in a product basis. Given your experience with numerical methods, you may recognize that for a state $∣ ψ ⟩ = \sum_{ij} C_{ij} ∣ i ⟩_{A} \otimes ∣ j ⟩_{B}$ , the Schmidt coefficients $λ_{i}$ are the singular values of the matrix $C$ .

3. Schmidt Decomposition of the Bell States

Let's apply this to the Bell states. Consider $∣ Φ^{+} ⟩ = \frac{1}{2} (∣00 ⟩ + ∣11 ⟩)$ . We can rewrite this as:

∣ Φ^{+} ⟩ = \frac{1}{2} ∣0 ⟩_{A} \otimes ∣0 ⟩_{B} + \frac{1}{2} ∣1 ⟩_{A} \otimes ∣1 ⟩_{B}

This expression is already in the form of a Schmidt decomposition!

The Schmidt coefficients are $λ_{1} = 1/ 2$ and $λ_{2} = 1/ 2$ .
The orthonormal basis for system A is ${∣ u_{1} ⟩_{A}, ∣ u_{2} ⟩_{A}} = {∣0 ⟩_{A}, ∣1 ⟩_{A}}$ .
The orthonormal basis for system B is ${∣ v_{1} ⟩_{B}, ∣ v_{2} ⟩_{B}} = {∣0 ⟩_{B}, ∣1 ⟩_{B}}$ .

The Schmidt rank is $k = 2$ , which is greater than 1. Therefore, $∣ Φ^{+} ⟩$ is an entangled state. The same logic applies to all four Bell states. For example, for $∣ Ψ^{-} ⟩ = \frac{1}{2} (∣01 ⟩ - ∣10 ⟩)$ :

∣ Ψ^{-} ⟩ = \frac{1}{2} ∣0 ⟩_{A} \otimes ∣1 ⟩_{B} - \frac{1}{2} ∣1 ⟩_{A} \otimes ∣0 ⟩_{B}

This is also a Schmidt decomposition with rank 2. The Schmidt coefficients are again $λ_{1} = 1/ 2$ and $λ_{2} = 1/ 2$ .

4. Quantifying Entanglement: Von Neumann Entropy

The Schmidt rank tells us whether a state is entangled, but it doesn't fully capture how entangled it is. For that, we turn to the von Neumann entropy.

In the previous lesson, you saw that for a pure bipartite state $∣ ψ ⟩_{A B}$ , the reduced density matrix of a subsystem, say $ρ_{A} = Tr_{B} (∣ ψ ⟩_{A B} ⟨ ψ ∣_{A B})$ , can represent a mixed state. The "mixedness" of this reduced state is a measure of the entanglement of the original pure state. The von Neumann entropy quantifies this mixedness.

Quantum entanglement | Random physics

The article 'Quantum entanglement' provides a concise mathematical definition of the Schmidt decomposition and the von Neumann entropy, along with a Python implementation.

Please read the sections 'Density matrix' (specifically the subsection 'Bipartite systems') and the definition of von Neumann entropy just below the code block in the first section. Focus on: How the Schmidt decomposition is expressed in terms of the matrix of amplitudes X (eq. |ψ⟩ = √P|uA⟩⊗|uB⟩). The definition of the von Neumann entropy S = −Tr(ρ log ρ). The Python function vn_ent which calculates this from the eigenvalues of the reduced density matrix.

As the resource explains, the von Neumann entropy of a state $ρ$ is:

S (ρ) = - Tr (ρ lo g_{2} ρ)

To calculate the entanglement entropy of a pure bipartite state $∣ ψ ⟩_{A B}$ , we compute the von Neumann entropy of one of its reduced density matrices, e.g., $S (ρ_{A})$ .

A crucial connection is that the eigenvalues of the reduced density matrix $ρ_{A}$ are precisely the squares of the Schmidt coefficients, $λ_{i}^{2}$ . Since the trace of an operator is the sum of its eigenvalues, the entropy can be calculated directly from the Schmidt coefficients:

S (ρ_{A}) = - i \sum λ_{i}^{2} lo g_{2} (λ_{i}^{2})

Let's calculate this for a Bell state, like $∣ Φ^{+} ⟩$ .

Find the Schmidt coefficients: We already found them: $λ_{1} = 1/ 2$ , $λ_{2} = 1/ 2$ .
Find the eigenvalues of $ρ_{A}$ : The eigenvalues are $λ_{1}^{2} = 1/2$ and $λ_{2}^{2} = 1/2$ .
Calculate the entropy: $S = - (\frac{1}{2} lo g_{2} (\frac{1}{2}) + \frac{1}{2} lo g_{2} (\frac{1}{2})) = - (\frac{1}{2} (- 1) + \frac{1}{2} (- 1)) = 1$

Alternatively, we can explicitly compute the reduced density matrix $ρ_{A}$ :

ρ_{A} = Tr_{B} (∣ Φ^{+} ⟩ ⟨ Φ^{+} ∣) = Tr_{B} [\frac{1}{2} (∣00 ⟩ ⟨ 00∣ + ∣00 ⟩ ⟨ 11∣ + ∣11 ⟩ ⟨ 00∣ + ∣11 ⟩ ⟨ 11∣)]

ρ_{A} = \frac{1}{2} (∣0 ⟩ ⟨ 0∣ ⟨ 0∣0 ⟩ + ∣0 ⟩ ⟨ 1∣ ⟨ 0∣1 ⟩ + ∣1 ⟩ ⟨ 0∣ ⟨ 1∣0 ⟩ + ∣1 ⟩ ⟨ 1∣ ⟨ 1∣1 ⟩)

ρ_{A} = \frac{1}{2} (∣0 ⟩ ⟨ 0∣ + ∣1 ⟩ ⟨ 1∣) = \frac{1}{2} I

This is the maximally mixed state for a single qubit. Its eigenvalues are $1/2$ and $1/2$ , which gives an entropy of 1, confirming our calculation.

For a $d$ -dimensional system (for a single qubit, $d = 2$ ), the maximum possible von Neumann entropy is $lo g_{2} (d)$ . Since the Bell states yield an entropy of $lo g_{2} (2) = 1$ , they contain the maximum possible amount of entanglement for a two-qubit system. This is why they are called maximally entangled states.

Conclusion

In this lesson, we delved into the quintessential quantum phenomenon of entanglement using the Bell states as our primary example.

Key Takeaways:

The four Bell states are a basis of maximally entangled two-qubit states, which can be generated by applying a Hadamard gate and a CNOT gate to the computational basis states.
The Schmidt decomposition provides a canonical form for any pure bipartite state, and its rank directly reveals whether the state is entangled ( $k > 1$ ) or separable ( $k = 1$ ).
The von Neumann entropy of a reduced density matrix, $S (ρ) = - Tr (ρ lo g_{2} ρ)$ , quantifies the entanglement in a pure bipartite state.
This entropy can be calculated directly from the Schmidt coefficients $λ_{i}$ as $S = - \sum_{i} λ_{i}^{2} lo g_{2} (λ_{i}^{2})$ .
The Bell states all have a Schmidt rank of 2 and a von Neumann entropy of 1, which is the maximum possible for a two-qubit system, confirming they are maximally entangled.

Preview of the Next Lesson:

We have now covered the fundamental building blocks of quantum circuits: single-qubit states and gates, two-qubit gates, measurement, and entanglement. In the final lesson of this introductory module, we will bring all these concepts together. You will implement quantum circuits in a simulator and verify their correctness by comparing measured state statistics with exact quantum predictions. This will be a practical, hands-on lesson to solidify your understanding before we move on to foundational quantum algorithms.